Linear matrix inequalities, Riccati equations, and indefinitestochastic linear quadratic controls
Rami, M.A.; Xun Yu Zhou
Automatic Control, IEEE Transactions on
Volume 45, Issue 6, Jun 2000 Page(s):1131 - 1143
Digital Object Identifier 10.1109/9.863597
Summary:This paper deals with an optimal stochastic linear-quadratic (LQ)
control problem in infinite time horizon, where the diffusion term in
dynamics depends on both the state and the control variables. In
contrast to the deterministic case, we allow the control and state
weighting matrices in the cost functional to be indefinite. This leads
to an indefinite LQ problem, which may still be well posed due to the
deep nature of uncertainty involved. The problem gives rise to a
stochastic algebraic Riccati equation (SARE), which is, however,
fundamentally different from the classical algebraic Riccati equation as
a result of the indefinite nature of the LQ problem. To analyze the
SARE, we introduce linear matrix inequalities (LMIs) whose feasibility
is shown to be equivalent to the solvability of the SARE. Moreover, we
develop a computational approach to the SARE via a semi-definite
programming associated with the LMIs. Finally, numerical experiments are
reported to illustrate the proposed approach
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